Mathematical Foundations¶

This document establishes the mathematical framework underlying quantitative finance and the Neutryx implementation.

Table of Contents¶

Probability Theory Foundations
Stochastic Processes
Stochastic Calculus
Risk-Neutral Pricing
Martingale Theory
Change of Measure

Probability Theory Foundations¶

Probability Space¶

All financial models are defined on a filtered probability space \((\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \geq 0}, \mathbb{P})\), where:

\(\Omega\) is the sample space (set of all possible outcomes)
\(\mathcal{F}\) is a \(\sigma\)-algebra of events
\(\{\mathcal{F}_t\}_{t \geq 0}\) is a filtration (increasing family of \(\sigma\)-algebras representing information flow)
\(\mathbb{P}\) is a probability measure

Reference: [Shreve, 2004], Chapter 1

Conditional Expectation¶

For a random variable \(X\) and \(\sigma\)-algebra \(\mathcal{G} \subseteq \mathcal{F}\), the conditional expectation \(\mathbb{E}[X | \mathcal{G}]\) is characterized by:

\(\mathbb{E}[X | \mathcal{G}]\) is \(\mathcal{G}\)-measurable
For all \(A \in \mathcal{G}\): \(\int_A \mathbb{E}[X | \mathcal{G}] \, d\mathbb{P} = \int_A X \, d\mathbb{P}\)

Properties: - Tower property: \(\mathbb{E}[\mathbb{E}[X | \mathcal{G}]] = \mathbb{E}[X]\) - Taking out what is known: \(\mathbb{E}[Y X | \mathcal{G}] = Y \mathbb{E}[X | \mathcal{G}]\) if \(Y\) is \(\mathcal{G}\)-measurable

Stochastic Processes¶

Brownian Motion¶

A Brownian motion (or Wiener process) \(\{W_t\}_{t \geq 0}\) is a continuous-time stochastic process with:

\(W_0 = 0\) almost surely
Independent increments: For \(0 \leq s < t\), \(W_t - W_s\) is independent of \(\mathcal{F}_s\)
Stationary increments: \(W_t - W_s \sim \mathcal{N}(0, t-s)\)
Continuous paths: \(t \mapsto W_t\) is continuous almost surely

Properties: - \(\mathbb{E}[W_t] = 0\), \(\text{Var}(W_t) = t\) - \(\text{Cov}(W_s, W_t) = \min(s, t)\) - Quadratic variation: \([W]_t = t\) (a.s.) - Non-differentiable: Paths are continuous but nowhere differentiable

Reference: [Shreve, 2004], Chapter 3; [Björk, 2009], Chapter 4

Implementation: All diffusion models in src/neutryx/models/ use Brownian motion as the fundamental noise source.

Geometric Brownian Motion¶

The geometric Brownian motion (GBM) is the classical model for stock prices:

\[ dS_t = \mu S_t \, dt + \sigma S_t \, dW_t \]

Solution (via Itô's lemma):

\[ S_t = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right) \]

Properties: - \(S_t > 0\) always (lognormal distribution) - \(\mathbb{E}[S_t] = S_0 e^{\mu t}\) - \(\text{Var}(S_t) = S_0^2 e^{2\mu t}(e^{\sigma^2 t} - 1)\) - Log-returns are normally distributed: \(\log(S_t/S_0) \sim \mathcal{N}((\mu - \sigma^2/2)t, \sigma^2 t)\)

Reference: [Shreve, 2004], Chapter 4; [Hull, 2022], Chapter 15

Implementation: Black-Scholes model in src/neutryx/models/bs.py

Poisson Process¶

A Poisson process \(\{N_t\}_{t \geq 0}\) with intensity \(\lambda > 0\) satisfies:

\(N_0 = 0\)
Independent increments
\(N_t - N_s \sim \text{Poisson}(\lambda(t-s))\) for \(s < t\)

Properties: - \(\mathbb{E}[N_t] = \text{Var}(N_t) = \lambda t\) - Inter-arrival times are exponentially distributed: \(\text{Exp}(\lambda)\) - Compensated Poisson process: \(\tilde{N}_t = N_t - \lambda t\) is a martingale

Application: Jump models (Merton, Kou, Variance Gamma)

Reference: [Cont & Tankov, 2004], Chapter 2

Implementation: Jump-diffusion models in src/neutryx/models/jump_diffusion.py, kou.py, variance_gamma.py

Lévy Processes¶

A Lévy process \(\{X_t\}_{t \geq 0}\) is a càdlàg process with:

\(X_0 = 0\)
Independent increments
Stationary increments
Stochastic continuity

Lévy-Khintchine representation: The characteristic function has the form:

\[ \mathbb{E}[e^{i u X_t}] = \exp(t \psi(u)) \]

where \(\psi(u)\) is the characteristic exponent:

\[ \psi(u) = i \gamma u - \frac{\sigma^2}{2} u^2 + \int_{\mathbb{R}} \left(e^{iux} - 1 - iux\mathbf{1}_{|x|<1}\right) \nu(dx) \]

\(\gamma \in \mathbb{R}\): drift
\(\sigma \geq 0\): Gaussian component
\(\nu\): Lévy measure (jump intensity)

Examples: - Brownian motion: \(\nu = 0\) - Poisson process: \(\nu = \lambda \delta_1\) (point mass at 1) - Variance Gamma: \(\nu(dx) = \frac{C}{|x|} e^{-M|x|} dx\) (infinite activity)

Reference: [Cont & Tankov, 2004], Chapters 3-4

Stochastic Calculus¶

Itô Integral¶

For a progressively measurable process \(\{X_t\}\) and Brownian motion \(\{W_t\}\), the Itô integral is:

\[ \int_0^t X_s \, dW_s \]

Properties: - Martingale property: \(\mathbb{E}\left[\int_0^t X_s \, dW_s\right] = 0\) - Isometry: \(\mathbb{E}\left[\left(\int_0^t X_s \, dW_s\right)^2\right] = \mathbb{E}\left[\int_0^t X_s^2 \, ds\right]\) - Not path-by-path: Defined as a limit in \(L^2(\mathbb{P})\)

Reference: [Shreve, 2004], Chapter 4; [Björk, 2009], Chapter 5

Itô's Lemma¶

Itô's Lemma is the chain rule for stochastic calculus. If \(X_t\) satisfies:

\[ dX_t = \mu(X_t, t) \, dt + \sigma(X_t, t) \, dW_t \]

and \(f(x, t) \in C^{2,1}\), then:

\[ df(X_t, t) = \left(\frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{\sigma^2}{2} \frac{\partial^2 f}{\partial x^2}\right) dt + \sigma \frac{\partial f}{\partial x} \, dW_t \]

Shorthand: \(dW_t^2 = dt\), \(dt \cdot dW_t = 0\), \(dt^2 = 0\)

Example: Deriving GBM solution

Given \(dS_t = \mu S_t dt + \sigma S_t dW_t\), let \(f(S, t) = \log S\):

\[ \begin{align} d\log S_t &= \frac{1}{S_t} dS_t - \frac{1}{2S_t^2} (dS_t)^2 \\ &= \frac{1}{S_t}(\mu S_t dt + \sigma S_t dW_t) - \frac{1}{2S_t^2} \sigma^2 S_t^2 dt \\ &= \left(\mu - \frac{\sigma^2}{2}\right) dt + \sigma dW_t \end{align} \]

Integrating: \(\log S_t = \log S_0 + (\mu - \sigma^2/2)t + \sigma W_t\)

Reference: [Shreve, 2004], Chapter 4; [Björk, 2009], Chapter 4

Implementation: Itô's lemma is used throughout for model derivations and is leveraged via JAX automatic differentiation.

Multidimensional Itô's Lemma¶

For \(\mathbf{X}_t \in \mathbb{R}^n\) satisfying:

\[ d\mathbf{X}_t = \boldsymbol{\mu}(\mathbf{X}_t, t) \, dt + \boldsymbol{\Sigma}(\mathbf{X}_t, t) \, d\mathbf{W}_t \]

where \(\mathbf{W}_t\) is a \(d\)-dimensional Brownian motion, and \(f(\mathbf{x}, t) \in C^{2,1}\):

\[ df(\mathbf{X}_t, t) = \frac{\partial f}{\partial t} dt + \sum_{i=1}^n \frac{\partial f}{\partial x_i} dX_t^i + \frac{1}{2} \sum_{i,j=1}^n \frac{\partial^2 f}{\partial x_i \partial x_j} dX_t^i dX_t^j \]

where \(dX_t^i dX_t^j = (\boldsymbol{\Sigma} \boldsymbol{\Sigma}^T)_{ij} dt\)

Application: Multi-asset derivatives, Heston model (2D system)

Reference: [Shreve, 2004], Chapter 4

Risk-Neutral Pricing¶

Fundamental Theorem of Asset Pricing¶

The First Fundamental Theorem states:

A market model is arbitrage-free if and only if there exists an equivalent martingale measure \(\mathbb{Q}\) (risk-neutral measure).

Under \(\mathbb{Q}\), discounted asset prices are martingales:

\[ \frac{S_t}{B_t} = \mathbb{E}^{\mathbb{Q}}\left[\frac{S_T}{B_T} \, \Big| \, \mathcal{F}_t\right] \]

where \(B_t = e^{\int_0^t r_s ds}\) is the money market account.

Second Fundamental Theorem: The market is complete if and only if the risk-neutral measure \(\mathbb{Q}\) is unique.

Reference: [Shreve, 2004], Chapter 5; [Björk, 2009], Chapter 10

Risk-Neutral Pricing Formula¶

The arbitrage-free price of a contingent claim \(V_T = g(S_T)\) at time \(t\) is:

\[ V_t = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}[g(S_T) | \mathcal{F}_t] \]

Interpretation: 1. Compute expected payoff under \(\mathbb{Q}\) (not real-world \(\mathbb{P}\)) 2. Discount at risk-free rate \(r\)

Example: European call option

\[ C(S_0, K, T) = e^{-rT} \mathbb{E}^{\mathbb{Q}}[\max(S_T - K, 0)] \]

Reference: [Hull, 2022], Chapter 13; [Shreve, 2004], Chapter 5

Implementation: All pricing engines in src/neutryx/engines/ implement risk-neutral pricing via Monte Carlo, PDE, or Fourier methods.

Girsanov Theorem¶

Girsanov's Theorem allows us to change the drift of a Brownian motion by changing the probability measure.

Let \(\{W_t\}\) be a Brownian motion under \(\mathbb{P}\). Define the Radon-Nikodym derivative:

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\Big|_{\mathcal{F}_T} = Z_T = \exp\left(-\int_0^T \theta_s \, dW_s - \frac{1}{2}\int_0^T \theta_s^2 \, ds\right) \]

Then under \(\mathbb{Q}\):

\[ \tilde{W}_t = W_t + \int_0^t \theta_s \, ds \]

is a Brownian motion.

Application: Deriving the risk-neutral measure from physical measure

Under \(\mathbb{P}\): \(dS_t = \mu S_t dt + \sigma S_t dW_t\)

Choose \(\theta_t = \frac{\mu - r}{\sigma}\) (market price of risk). Then under \(\mathbb{Q}\):

\[ dS_t = r S_t dt + \sigma S_t d\tilde{W}_t \]

Reference: [Shreve, 2004], Chapter 5; [Björk, 2009], Chapter 11

Martingale Theory¶

Martingales¶

A process \(\{M_t\}\) adapted to \(\{\mathcal{F}_t\}\) is a martingale under \(\mathbb{P}\) if:

\(\mathbb{E}[|M_t|] < \infty\) for all \(t\)
\(\mathbb{E}[M_t | \mathcal{F}_s] = M_s\) for all \(s \leq t\)

Interpretation: Fair game - the expected future value equals the current value given current information.

Examples: - Brownian motion \(W_t\) is a martingale - \(W_t^2 - t\) is a martingale (compensated quadratic variation) - Discounted stock price under \(\mathbb{Q}\): \(e^{-rt} S_t\)

Reference: [Shreve, 2004], Chapter 3; [Björk, 2009], Chapter 6

Doob's Optional Stopping Theorem¶

For a martingale \(\{M_t\}\) and stopping time \(\tau\) with \(\mathbb{E}[\tau] < \infty\):

\[ \mathbb{E}[M_\tau] = \mathbb{E}[M_0] \]

Application: Pricing American options (optimal stopping problems)

Reference: [Shreve, 2004], Chapter 3

Change of Measure¶

Change of Numeraire¶

The change of numeraire technique allows pricing in different units.

Let \(N_t\) be a strictly positive traded asset (numeraire). Under the \(N\)-forward measure \(\mathbb{Q}^N\):

\[ \frac{S_t}{N_t} = \mathbb{E}^{\mathbb{Q}^N}\left[\frac{S_T}{N_T} \, \Big| \, \mathcal{F}_t\right] \]

Common numeraires: - Money market account: \(N_t = B_t = e^{rt}\) → standard risk-neutral measure \(\mathbb{Q}\) - Zero-coupon bond: \(N_t = P(t, T)\) → \(T\)-forward measure \(\mathbb{Q}^T\) - Stock price: \(N_t = S_t\) → stock measure (useful for volatility derivatives)

Example: Black's formula for bond options

Using the \(T\)-forward measure simplifies pricing: forward rates are martingales.

Reference: [Shreve, 2004], Chapter 6; [Brigo & Mercurio, 2006], Chapter 2

Implementation: Used implicitly in interest rate models (src/neutryx/models/vasicek.py, hull_white.py)

Cameron-Martin-Girsanov Formula¶

The general form of measure change (for continuous semimartingales):

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\Big|_{\mathcal{F}_T} = \exp\left(\int_0^T \theta_s \, dX_s - \frac{1}{2}\int_0^T \theta_s^2 \, d[X]_s\right) \]

where \([X]_s\) is the quadratic variation of \(X\).

Reference: [Shreve, 2004], Chapter 5

Feynman-Kac Formula¶

The Feynman-Kac theorem connects PDEs and expectations, providing the foundation for both PDE and Monte Carlo pricing.

Theorem: Suppose \(X_t\) satisfies:

\[ dX_t = \mu(X_t, t) \, dt + \sigma(X_t, t) \, dW_t^{\mathbb{Q}} \]

and define:

\[ u(x, t) = \mathbb{E}^{\mathbb{Q}}\left[e^{-\int_t^T r(X_s, s) ds} g(X_T) \, \Big| \, X_t = x\right] \]

Then \(u(x, t)\) satisfies the PDE:

\[ \frac{\partial u}{\partial t} + \mu(x, t) \frac{\partial u}{\partial x} + \frac{\sigma^2(x, t)}{2} \frac{\partial^2 u}{\partial x^2} - r(x, t) u = 0 \]

with terminal condition \(u(x, T) = g(x)\).

Black-Scholes PDE: For GBM under \(\mathbb{Q}\) (\(\mu = r\)):

\[ \frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{\sigma^2 S^2}{2} \frac{\partial^2 V}{\partial S^2} - r V = 0 \]

Reference: [Shreve, 2004], Chapter 5; [Björk, 2009], Chapter 7

Implementation: - Monte Carlo: Directly samples the expectation (src/neutryx/engines/mc.py) - PDE: Discretizes and solves the PDE (src/neutryx/models/pde.py)

Numerical Implementation Notes¶

JAX and Automatic Differentiation¶

Neutryx leverages JAX for automatic differentiation, which computes derivatives via:

Forward mode: Efficient for \(\mathbb{R}^n \to \mathbb{R}^m\) with \(n \ll m\)
Reverse mode (backpropagation): Efficient for \(\mathbb{R}^n \to \mathbb{R}^m\) with \(m \ll n\)

Greeks computation: \(\frac{\partial V}{\partial S}\), \(\frac{\partial^2 V}{\partial S^2}\) computed exactly via autodiff

Reference: [Bradbury et al., 2018]

Implementation: All models support automatic Greeks via JAX: src/neutryx/valuations/greeks/

Precision and Stability¶

Float32 vs Float64: Configurable precision (GPU prefers float32, CPU can use float64)
Log-space computations: Avoid overflow in characteristic functions
Variance reduction: Reduces simulation noise, improving convergence

Implementation: Precision set globally; variance reduction in src/neutryx/engines/variance_reduction.py

Summary¶

This mathematical foundation provides:

Probability framework: Filtered probability spaces, conditional expectation
Stochastic processes: Brownian motion, Lévy processes, jump processes
Stochastic calculus: Itô's lemma, quadratic variation
Risk-neutral pricing: Fundamental theorems, martingale measures
Measure changes: Girsanov theorem, change of numeraire
Feynman-Kac: PDE-expectation duality

All Neutryx models and numerical methods are built on these rigorous mathematical foundations, as detailed in References.

Next: Pricing Models Theory | Numerical Methods Theory